I can't believe I'm replying to spam, but...

Having been brought up in London, not very far from Heathrow, and even though I now live elsewhere in England I still work in North London, go to Chelsea quite often (see Team name) and travel about the city a fair bit, don't for God's sake do any of this unless you have money to burn.
I mean, cabs are good, and the door-to-door aspect is definitely the best thing about this, but the Underground goes right into Heathrow and breaks the back of these journeys at 10-20% of the price.

Heathrow to Chelsea: Piccadilly Line direct to Earls Court, then get a cab from there
Heathrow to West Ham: Elizabeth Line direct to Stratford, then get a cab from there
Heathrow to Cardiff: Heathrow Express to Paddington, change for Paddington to Cardiff, then get a cab from there [£65 not £300]
Heathrow to Manchester: Heathrow Express to Paddington, change for Paddington to Euston Square, change for London Euston to Manchester, then get a cab from there [£110 not £500]

Useful links
https://www.heathrowexpress.com/ - out of Heathrow to Paddington
https://tfl.gov.uk/plan-a-journey/ - within London
https://www.thetrainline.com/ - to other cities, can combine all trains on one ticket